Higher order derivatives of matrix functions

TL;DR

This paper develops a comprehensive theory for higher order derivatives of matrix functions, extending previous results and connecting to quantum perturbation theory, with practical numerical methods and examples.

Contribution

It introduces a block upper triangular form for higher order derivatives of matrix functions, generalizes Daleckind-Kren formulas, and links to quantum perturbation theory and complex step methods.

Findings

Derived conditions for existence of higher order derivatives.

Established a generalized Daleckind-Kren formula.

Connected derivatives to quantum perturbation theory and numerical approximations.

Abstract

We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610-620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Dalecki\u{i}-Kre\u{i}n formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Fr\'echet derivatives. Block forms of complex step approximations are introduced and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples.